Functions, domain and range

6 minAS Level1.2.11.2.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can use the words function, domain and range the way examiners do, read Cambridge function notation, and find the range of a simple function.

A vending machine that never glitches

Press button B4 on a vending machine and you get crisps. Every time.

That reliability is the whole idea of a function: each input gives exactly one output.

Two buttons giving the same snack? Fine. Still a function.
One button dropping crisps on Monday and a cola on Tuesday? Not a function.

Where the analogy breaks: a real machine runs out of crisps. A function never does, and its list of buttons can be infinite.

Definition

Function

A function is a rule that maps each element of a set of inputs, the domain, to exactly one output value.

Three names for how inputs and outputs can pair up:

one-one: each output comes from just one input (a function);
many-one: two or more inputs share an output (still a function);
one-many: one input has several outputs (not a function).

Many-one is allowed (two inputs can share an output). One-many is not: an input with two outputs breaks the "exactly one" rule.

Reading the notation

Cambridge writes the same function in two ways, and you need both:

Notation	Read it as
$f(x) = 3x - 1$	"f of $x$ equals $3x - 1$ "
$f : x \mapsto 3x - 1$	"f maps $x$ to $3x - 1$ "
$\text{for } x \in \mathbb{R}$	"for every real number $x$ "
$\text{for } x \geqslant 2$	"the domain is $x \geqslant 2$ only"

Two symbols in that table:

$\in$ means "is a member of" or "belongs to".
$\mathbb{R}$ is the set of all real numbers: every number on the number line, including negatives, fractions, and numbers like $\sqrt2$ whose decimals never end.

So " $x \in \mathbb{R}$ " just means " $x$ can be any real number".

The domain is part of the function. These two are different functions, even though the rule is the same:

$f(x) = x^2$ for $x \in \mathbb{R}$
$f(x) = x^2$ for $x \geqslant 0$

Definition

Domain and range

The domain of a function is the set of input values it is defined for.

The range is the set of all output values $f(x)$ that the function produces from its domain.

(A set is just a collection of numbers.)

Example: $f(x) = 2x$ for $x \in \{1, 2, 3\}$ .

Domain: $\{1, 2, 3\}$ , the allowed inputs.
Range: $\{2, 4, 6\}$ , the outputs you actually get.

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 2 questions above to continue.